Budapest is famous for all the termal baths. At my birthday we went to Szechenyi Termal Bath, a big complex with lots of indoor and outdoor pools and stayed there for hours. Really recommendable. The advantage in January is that there are no crowds in the city - Restaurants, Cafes etc. - all empty. Without reservations we could go to the nicest restaurants during the weekend. Especially on one of the ships on the Danube the food was very good.
Of course I took my photo equipment and the tripod was useful. During the day the weather was not that good, the last day it even snowed quite a lot. But in the evening it got very colorful with all the lights, especially with the world famous Chain Bridge, the Castle and Parliament.
Also nice was a walk over the big market (indoor) which is a little touristy but still there were lots of locals around to buy meat or vegetables.
Altogether we stayed for 3 nights which was just enough for a visit.
Enjoy the pictures.